In [1]}, p. 8, Shulman mentions that “a notion corresponding
roughly to the Chu construction was already introduced by (Bishop and Bridges in) [2]}
under the name complemented subset”. Actually, the notion of a complemented subset is already introduced by
Bishop in [3]}, pp. 66-69, under the name complemented set.
Here we explain in what sense Bishop's notion corresponds to the Chu construction. Namely, there is a Chu representation of
the thin category \(P̏^{\rrbracket \llbracket }(X)\)  of complemented subsets of a set \(X\)  into \(\textnormal {\textbf {Chu}}(\mathrm {\mathbf {Set}}, X \times X)\) . Notice that Bishop's motivation for introducing complemented (sub)sets is rooted
to his need to overcome problems generated by the use of negation in basic set and measure theory in a constructive setting (see [4]}, chapter 7, and [5]}). Hence, the connection described here between the Chu construction and Bishop's notion of complemented subsets seems to be accidental.
